CANADIAN BOARD OF EXAMINERS FOR PROFESSIONAL SURVEYORS
ATLANTIC PROVINCES BOARD OF EXAMINERS FOR LAND SURVEYORS
**________________________________________________________________________ **

**SCHEDULE I / ITEM 2** **March 1007 **

**LEAST SQUARES ESTIMATION AND DATA ANALYSIS **

**Note: This examination consists of 8 questions on 3 pages. ** **Marks **

**Q. No**** ** Time: 3 hours ** **** Value Earned **

1

Given the following mathematical models

2 2

1 1

0 ) ,

(

0 ) , (

2 1 2 2

1 1 1

*x*
*x*

**C**
** **
**C**
** **
**x**

**,**
**x**
**f**

**C**
** **
**C**
** **
**x**

**f**

l l

l l

=

=

=

=

=

=

=

=

where f_{1}andf_{2} are vectors of mathematical models, x_{1}and x_{2} are
vectors of unknown parameters, l_{1}andl_{2} are vectors of
observations,

2 1

2

1 ,C ,C and C

Cl l _{x}* _{x}* are covariance matrices.

a) Linearize the mathematical models b) Formulate the variation function

10

2

Given n independent measurements of a single quantity: l_{1},l_{2},...,l_{n}
and their corresponding standard deviations:σ_{1},σ_{2},...,σ_{n}, derive the
expression for the variance of the mean value using the Law of

Propagation of Variances. ^{10 }

3

Given the following variance-covariance matrix from a least squares adjustment for the coordinates of point A and B,

2

x m

5 2 1 1

2 5 1 2

1 1 10 5

1 2 5 10 C

= where x=[x_{A} y_{A} x_{B} y_{B}]^{T}.

a) Compute the correlation coefficient between x_{A} and y_{B}.
b) Compute the correlation coefficient between y_{A} and x_{B}.

c) Compute the variance-covariance matrix for the coordinate differences

A B x x

x= −

∆ and ∆y=y_{B} −y_{A}.

10

4

Given the covariance matrix of the horizontal coordinates (x, y) of a survey station, determine the semi-major, semi-minor axis and the orientation of the standard error ellipse associated with this station.

_{x} m^{2}
0196
.
0
0246
.
0

0246 . 0 0484 .

C 0 _{}

=

10

5

Define or explain the following terms:

1) Precision 2) Accuracy

3) Statistical independence 4) Type I error

5) Type II error 6) Expectation

7) Degree of freedom of a linear system

15

6

Given 20 independent measurements of a distance made with the same standard deviation of 0.033m and the calculated mean value µ= 37.615m, construct a 95% confidence interval for the population mean.

**Percentiles of Standard Normal Distribution **

α 0.002 0.003 0.004 0.005 0.01 0.025 0.05 Kα 2.88 2.75 2.65 2.58 2.33 1.96 1.64

where Kα is determined by the equation α= e dx 2

1 _{x} _{/}_{2}

K

− 2

∫∞

α π .

10

7

A baseline of calibrated length (µ) 1153.00m is measured 5 times independently with the same precision. The sample mean ( x ) and sample standard deviation (s) are calculated from the measurements:

x =1153.39m s = 0.06m

Test at a 90% confidence level (α) if the measured distance is significantly different from the calibrated distance.

**Percentiles of t distribution **
tα

Degree of

freedom ^{0}^{.}^{90}

t t_{0}_{.}_{95} t_{0}_{.}_{975} t_{0}_{.}_{99}

1 3.08 6.31 12.7 31.8

2 1.89 2.92 4.30 6.96

3 1.64 2.35 3.18 4.54

4 1.53 2.13 2.78 3.75

5 1.48 2.01 2.57 3.36

10

8

Given in the following is a diagram of a leveling network to be adjusted.

Table 1 contains all of the height difference measurements ∆*h** _{ij}*
necessary to carry out the adjustment. Assume that all observations have
the same standard deviation

*h**ij*

σ∆ = 0.04m. P1 is a fixed point with known elevations of H1 = 101.00m. Perform least squares adjustment on the leveling network to compute:

a) the adjusted elevations of point P2, P3 and P4 and their variance- covariance matrix;

b) the observation residuals;

c) the adjusted height differences; and
d) the a-posteriori variance factor σˆ_{0}.
**Table 1 **

**No. ** **i ** **j ** ∆*h** _{ij}*(unit: metre)

1 P1 P3 6.15

2 P1 P4 12.54

3 P3 P4 6.46

4 P1 P2 1.11

5 P2 P4 11.55

6 P2 P3 5.10

25

**Total Marks: ** 100
**P1 **

**P2 **

**P3 **
**P4 **